Intermeshing gear and pinion, and art or method of generating the same



Feb. 19, 1924; 1,483,899

C. H. LOGUE INTERMESHING GEAR AND PINION, AND ART OR METHOD OF GENERATING THE SAME FiledSepc. 3O 1919 2 Sheets-Sheet 1 ATTORNEYS. I

I INTERMESHING GEAR AND PINION, AND ART OR METHOD OF GENERATING THE SAME Feb. 19, I924; 1,483,899

(3. H. LOGUE Fileq Sept 30 1919 2 Sheets-Sheet Ibo A TTORNEYS.

Patented Feb. 19, 1924.

UNITED STATES 1 1,483,899 PATENT OFFICE.

CHARLES H. LOGUE, OF SYRACUSE, NEW YORK, ASSIGNOR TO BROWN-LIPE-CHAPIN 60., OF SYRACUSE, NEW YORK, A CORPORATION OF, NEW YORK.

INTERMESHING GEAR AND PINION, AND ART R METHOD OF GENERATING THE SAME.

Application tiled September 80, 1919. Serial No. 327,379.

To all whom it may concern:

Be it known that I, CHARLES H. Loom), a

citizen of the United States, residing at Syracuse, in the county of Onondaga and State of New York, have invented a oertaln new and useful Intermeshing Gear and Pinion, and Art or Method of Generating the Same, of which the following is a specification.

This invention relates to the art of generatin gear teeth and has for its object a. metho of generating or determining the tooth parts of gearsor pairs of intermeshing gears and pinions in which the teeth are I; so constructed that they are the strongest possible for the pitch diameter used, and the invention consists in the method, and in the gears embodying the novel features and characteristics, all as hereinafter set forth :0 and claimed.

In describing this invention, reference is had to the accompanying drawings in which like characters designate corresponding parts in all the views.

as Figure 1 is a diagrammatic view of a gear and pinion. lay out.

Figure 2 is a diagrammatic view illustrating the angles of action. I

Figure 3 is a diagrammatic view illusso trating the side angle.

' Figure 4 is a diagrammatic view showing that tooth dimensions vary directly as the pitch radius, for fixed values of gear ratio and number of teeth, increases.

This invention comprises a method of gencrating or determining the tooth parts or pairs of gears and pinions and the gears and pinions so produced in which the dimensions of the tooth parts are based upon the angles of approach and recess.

Heretofore, the usual procedure in designing gears has been to assume the pitch or the number of teeth then obtain gear and pinion tooth parts by means of certain proportionments based upon the desired pitch.

In gears embodying my invention the di, mensions of the tooth parts are based upon the angles of approach and recess.

The number of teeth in theoretical contact is assumed and, then the number of teeth; in the gear and pinion determined,

so that instead of assuming the pitch or the the resultant.

number of teeth and studying action or judging the action from general or commonly accepted ideas as to the most the in theoretical contact is also that which experlence and calculation have shown to be the most desirable for the gear ratio and conditions and as one skilled in the'art can readily determine the proper number, it is thought unnecessary to set forth herein any formula for obtaining the number of teeth in theoretical contact. The number of teeth in the gear and the pinion are determined from the number of teeth in contact.

The angle of approach and the angle of recess are determmed independently and have no direct connection with the angle of obli uity, the ratio of reduction (R) and asic angle, being the basis for this calculation. v

Having determined the angles of action,

the corresponding tooth parts are found.

The angle of approach is a function of the gear a dendum and the angle of recess a function of the inion addendum.

From the ang es of approach and recess the corresponding gear and pinion addenda are found in accordance with the following formulas 1 d see E sine p gear addendum. 7

P cos p M sec E In both formulas:

Tan ten of normal obliquity x se'eant spiral angle.

Cot E Jratio of reduction forspurgears.

Cot E ratio of reduction f 0r bevel gears.

tan g sme see it d=pitch radius of spur or virtual radius of bevel pinion.

= addendum.

P. cannot exceed value of p.

S2=basic angle (see explanation).

In the drawings, Fig. 1, D :the'pitch radius of the gear.

9 l =pitch angle of gear.

E zpitch angle of pinion.

In Fig. 2 the angles of action are illustrated- P.=angle of a roach.

p =angle of o lquity.

a :virtual obliquity angle of recess.

0 :angle of recess.

'A zpinion addendum B :the side angle.

In Fig. 4, the letters a, b, c designate respectively the position of the pinion tooth at beginning of action, the position at the end of the angle of approach, and the pinion tooth at the end of its action.

q; :angle of cutter or cutterobliquity.

A :s lral angle.

P =v1rtual obliquity.

3o P =angle of approach.

' v :angle of recess.

B :see diagram Fig. 4, side angle. I a :virtual obliquity recess angle. E zpitch angle of pinion.

r =ratio d :vittuaf pitch radius.

(1 =pitch radius.

40 A =pinion addendum.

*A zgear addendum.

N zteeth in contact. I

n :nnmber of teeth in pinion n :virtual number of teeth in pinion.

Sec E" 1 ratio 2 0 :basic angle.

The basic angle- 0) upon which compara- FO tive action is base is assumed, the general idea being that the percentage of rolling action durin each contact is-decreased directly with t 's angle.

This-basic" angle may be anything, for

example, 21--31', 2( )0C,"180'. In the design of bevel gears the following forinulas'are used. v

Virtual obliquity tan P=tan sec A Ratio of virtual pitch radius to pinion addendum ,Xm +1. v

Pinion addendum p d sec E 7 Angle of recess cos P 0=a-P inwhich cos a= Angle of approach Gear add when P is less than P. A, sec E d sine p )(sec E d sine p 0.05)

Teeth in contactassumcd 1.25 cos A minimum N =1,4c0s 1 5() cos A maxi Teeth in pinion when P. is less than P 360 N cos E P. 6B tan'a 360 n sec E Teeth in pinion when P. Pn: 6.2832; N cos E Teeth in contact when i o tan a n 6.2832 cos E For spur and bevel gears the same formulas are used except that in designing spur tooth dimensions vary directly as the pitch radius and the actions in all three gears there illustrated are based upon the same angles of approach and recess and that the tooth dimensions vary directly.as the pitch radii d, d, d.

Gears so designed will always result in the strongest possible tooth for .the itch diameters .on which the design is base and aside from checking the strength and the matter of -securing proportionate tooth thickness the pitch is found merely as a matter of curiosity-1 In gears constructed in accordance with my invention the. number of teeth are reducad to a'minimum.

The pitch is made as coarse as -possible; the face as narrow as possible, the proportioning of the tooth parts is not from the pitch. In fact, gears made in accordance with my invention are contrary to practically all the accepted theories upon which the subject of tooth proportion have heretofore been based.

In constructing and designing both straight and spiral tooth beveled gears we begin with assumed pitch diameters 2D and 211. From these diameters we determine the virtual radii D and al and consider these radii as the pitch radii as for spur 16 gears upon which the tooth action is developed.

All dimensions calculated upon these radii D and d are referred to as virtual parts, as the virtual number of teeth would 20 be the number of teeth contained in the virtual diameters 2D and 2th, on a basis of the virtual pitch.- Virtual obliquit is therefore the real or acting obliquity of an action. Reference to basic values apply to the engagement of the pinion with the crown gear or rack.

With bevel gears such as used for automobile drive, there is little danger of encountering inefiicient action by making the gear addendum too long. Therefore if full effective length is desired which results in an angle of approach equalling the angle of obliquity in accordance with formula 6 hereina ter given.

For 'the bevel gear ingeneral, it will be necessary to guard against an undersirable sliding-action of the angle of approach as well as upon the recess. This will" be neces- B=e-5 inwhich 57.296a: T d,

x sinee$= v The formula heretofore given for the gear to and pinion addenda are derived as follows DESIGN or never. (mans. Tooth parts.

Virtual obliquity Tan P =tan see A (1) Ratio of virtual pitch radius to pinion addendum sec E r cos P (2) cos (PHI) Pinion addendum I A,= or (3) Angle of recess If P. exceeds P, make P.=P

, Gear addendum when .P, =P,

Gear add. when P'. is less than P,

A,='sec E d sine p, or see E,d, sine p (6) AF sec E, ,d sin? 1%) (sec E,d sine P 0.05 1

Teeth in contacb-asumed N 1:4 cos A 1,25 cos A minimum 1.50 cos A maximum Teeth in pinion when P.= P

6.2832 N cos E n tan 0:

Teeth contact when P,.=P I

tan a n 6.2832 cos angle of cutter. P =virtual obliquity. P. angle of approach. =angle of recess. B =see diagram Fig. 3. oc t=virtual obliquity and recess L. A =spiral angle. E, pitch angle of pinion.

tior ra A Q=2133' when quietest ossible operation is desiredcase hardene gears.

(E -20 0 represents more conservative designcase hardened gears. Q=15--0' recommended for general purposes-untreated materials.

As this angle is increased, theoretlcal wear increases and noise of operation is reduced." For case hardened ears however the higher angle (2133') al ows a greater depth of caseoonsequently harder wearing surfaces, which will more than overcome theoretical wear losses.

The basic angle (Q) might be defined as the maximum angle of recess (6) for an infinity co ratio, that is, for a pinion engaged with a rack.

According to the formulas herein introduced, both the angles of approach and of recess are constant or a given gear ratio for each assumed basic angle (9).

The angle (21 33) is recommended when the quietest possible 0 eration is desired,'

if necessary at a sacri ce off wearing qualities. An angle of 15-0 is recommended for eneral pur oses and for untreated materials ut it will found that the estimated loss 'due to increased sliding action through the use of the higher basic angle will. be more than overcome by the ability to then em- (n dii E ing surface as the pitch is increased, which is the result of the higher angle.

Lower values for the angle [5 may be em ployed as desired, but 21 -33 expresses the high limit of good practice for case hardened steels, and 18 0 the high limit for untreated materials. The basic angle represents the angle of recess (6) for an infinity ratio, that is, for a pinion engaged in a rack.

The assumption as to normal obliquity (cp) (cutter angle) is a matter of judgment. In general the 20 degree is recommended. The 14% degree is not satisfactory for high duty and cannot compare, either as to strength or wearing quality, with the 20 degree.

In straight tooth gears the normal obliquity (cutter angle) and virtual obliquity are equal, that is, the obliquity of the cutter used in the acting obliquity of the gears.

For spiral gears, however, the virtual obliquity varies with the s iral angle, the

normal obliquity being fixe This relation is shown by formula 1 above.

The tooth action of spiral bevel gears is exactly that of the straight tooth bevel gear; that is, any one section may be considered as such. The normalobliquity need not be considered in' laying out or calculating tooth parts for such gears. After the virtual obliquity is determined by means of formula 1, the normal obliquity is of no further interest.

Normal: obliquity is here referred to as cutter angle-to avoid confusion with the acting (virtual) obliquity.

To avoid interference, the pinion dedendum must either be suificiently deep or sufficiently shallow. The gear addendum must either clear the flank of pinion tooth, that is, avoid all contact below base line, or be cut ofl:' so short that interference is impossible, as per the following formula PINION cinnamon.

To avoid interference (1 cos (b) (sine @035) sec E (or see E) Minimum pinion dedendum. Virtual pitch radius of pinion; Pitch angle of inion.

Normal obliquity of action.

gree ters.

includes the general run of bevel drive gears.

Pinion dedendum may be deeper, to accommodate gear addendum but it must be made no shallower unless the gear addendum is made so short that it does not overlap the base circle of the pinion on the line of cen- In such cases the gear addendum equals d,(cos q: 03,) and the corresponding pinion dedendum (d equals:

(Z [d,' (cos d,)]+bottom clearance.

The dedendum of pinion is of extreme importance-it must either exceed the value 05 or fall short of d,,. Dimensions between these values will cause interference. The value d,, is not practical for the general run of drive bevel gears as the resultant loss of action on the angle of approach will cause discontinuity of action unless the pinion addendum is made excessively long, causing maximum wear.

Sec E 1 for bevel gears.

ratio.

ratio gears produced by a generating process, 0 which a rack or crown gear is the base, interference is entirely a matter of pinion dedendum.

Interference has no direct connection with either numbers of teeth or pitch; it is a question of pitch diameters and obliquity of action.

Formula 6 above gives the full effective length of gear addendum, that is, an addendum resulting from an angle of approach, equalling the angle of obliquity; This is the length to which the gear addendum must be restricted in order .to avoid interference.

-An excellent way to encounter interference is to make the gear addendum sli htly shorter than this length when a me or crown gear is to be engaged. Since the usual automobile drive gear has practically a crown gear formation, it is reconfinended, in order to avoid interference, in straight Sec E= 1 for spur gears.

tooth ears especially, that we make the For spiral bevel gears, or for helical spur gears, multiply these values by the cosine of the spiral or helix angle, that is N =1A0 cos. A.

The number of teeth in'contact increases directly as the number of teeth only when the ratio ofthe virtual" pitch radius to pinion addendum (r) (remains constant, and as a result the number of teeth in contact, for any tooth proportion based upon the pitch, increases but slowly as compared with the increase in the number of teeth. Therefore noise of operation then increases as we increase the number of teeth, altho not quite in direct proportion and increases directly with the number of teeth when N is assumed.

Based upon the number of teeth incontact (N and the angles of action (P 0 and B) we now determine the number of teeth in pinion by formula 9 or when the angle of approach equals the obliquity by formula 10, without the necessity of calculatin angle.

The number of teeth found wi usually be a fractional number. The nearest integral value must be taken, kee ing within the limits as expressed in formu a 8 for the minimum and maximum number of teeth incontact. Check this by means of second part of formula 9 or when approach equals an 1e of virtual obliquity, by formula 11.

t may sometimes be necessary to make a slight readjustment of pitch diameters to agree with the ratio of the numbers of teeth thus determined. 1 N

The number of teeth in gear is of course the number of teeth in inion times the ratio of reduction (N==nR The usual practice is to hold the spiral angle between 30 0' and 35 0. Angles below thirty resulting in a product little better than straight tooth gears and angles above thirty-five avoided on account of the resultant axial thrust.

It might be suggested that we increase the angle to 45 0' or even higher andbuild ax e housings sufiiciently sturdy to carry the additional thrust. This would allow the use of extremely low ratios and bring the spiral bevel into direct competition with the worm gear. Construction ca able of carrying worm gearin will certain y properly support a spiral vel set of extreme angle. I J

t might be well to note here that emlo ing a 12" cutter,. as is standard pracem loying sine P, we make this value P y 0.0 0 for 141 and 0.120 for 20obliquity.

After determining the angles of. action P, and 6 and thereby the addendum of gear and. of pinion, the next step is to assume a theoretical number of teeth in contact which will assure continuous contact.

It has been found from practice that the most efiicient value for this is in the neighborhood of 1.40 and that it should be held between 1.25 and 1.50.

5" pitch cone distance is 56 26'.

,tice in cutting s iral bevel ears, the nearer the sine of spira angle equa s average p. 0. distance i thecloser we approximate the true spiral. Then a spiral angle of 30 is best suited for a at with an average pitch cone distance of 3" and the proper spiral angle (Agxfonlai be seen therefore that better gears may be produced by raising the spiral angle, as the avera automobile drive gear has a pitch cone distance of 5.5.

Reference to Fig. 4 illustrates that for a given value of r the strength of the pinion will var directly as the average virtual pitch ra ius (d,) for a given virtual number of teeth.

Ingeneral, the points to keep in mind regardmg bevel gear design are as follows': Reduce the number of teeth to a m1n1- mum.

Make the pitch as coarse as possible.

Make the face as narrow as is possible.

Proportion pinion dedendum to avoid interference.

Restrict the angle of action to avoid undue friction.

Avoid proportioning the tooth part from the pitch.

Avoid a long pinion addendum.

Avoid a short pinion dedendum.

Avoid a large number of teeth in pinion.

Avoid a fine pitch.

Avoid a long are of recess.

Avoid a. wide face.

Avoid a large number of teeth in theoretical' contact.

In fact, go contrary to practically all of the accepted theories upon the subject of tooth PIOPOItiOIItO produce quiet efiicient gears.

The following example has been worked 5 out in strict. accordance with the foregoing s cification in order to supply an explanation of the principles involved and to assist in their application. I

We are usually supplied with the pitch radius of the gear and the ratio of reduction. In this example we have:

Pitch radius of gear, D 5.3" Ratio of reduction, R 7.0

46 The pitch radius of pini C'utter obliquity.

The first step is to decide upon a cutter obliquity D). In general, degrees is recommended but this assumption is limited only by the angle of the cutters available. In this example 20 degrees is used.

ANGLE OF SPIRAL.

The angle of spiral should not be less than degrees and may be made as high as desired, within practical limits. Too

I high an angle, however, will force the number of teeth in contact,

N 1.40 cos A and if we assume, say 1.05 as a basis for the spiral angle, we have:

1.05 i COS A m therefore, A:41 24". Let us assume, however, that in this particular example an angle not over 0 is desired, on account of spiral thrust limitations.

. Virtual obliquity.

The first calculation necessary is to findthe virtual obliquity (P) by formula 1, we have: 1

Tan P= tanr= sec A= 0.3640 x 1.3054 o.475o==25 25'.

Basic angle. I We must now assume a basic angle (Q) 95 upon which comparative action is based; the

' general idea being that the percentage of rolling action during each contact is decreased directly with this angle although this relation has not yet been definitely established.

10 This angle may be made as desired, al-

though certain recommendations are made as to the proper limits. For extreme ratios it is best to take the maximum recommendation that is 21 33'. This is especially de- 105 sirable in cases where the pinion diameter is small and the coarsest possible pitch is required. It W111 be found that a theoretical loss due to the use of our maximum recommendation is more than made up by our ability to then apply a deeper case to the tooth surface. I This recommendation (21 33') should not be exceeded as extreme wear begins just beyond this point.

For general practice a basic angle of 20 0 isrecommended and for untreated materials 18 0' is good practice. Our present example being rather extreme and the pinion diameter; small, let Q=21 33.

Angle of approach.

The angle of approach and angle of recase are determined independently and have basic angle.

no direct connection with the angle of obliquity: the ratio of reduction (R) and the basic angle (9) being the basis of their calsec E, 1.0102

in which, E =8 8", sec E 210102, tan 9:

0.3949 angle of approach may be found di- 1 rect by referring to Fig. 5.

Angle of recess.

The angle of recess is found by means of formula 2. In this formula the ratio of the pinion addendum to its virtual pitch radius is employed. While this ratio (1) will change with each obliquity, it will be found that the resultant angle of recess is con.- stant for a given ratio of reduction and The advantage in employing the ratio (1") being that the pinion addendum may then be found by a simple calculation, see formula 3.

For a determlnation of 1' we have:

Cos P=c0s 25 25'=0.9032.

Cos (P+0=cos 4.6 58=0.6824.

According to formula 2,

sec P 1.0203 B f e992 1 cos (P+S2) 0.6824 7 a For the angle of recess (6)) by formula l we have: I

COEiP 0.9032

Cos 0.6482 or 46 50';

Having the desired angles of action we now proceed to find corresponding tooth parts by means of formulae 3 and 6, or 7, as the case may be. The angle of approach is a function of the ge'ar addendum and the angle of recess a function of the pinion addendum.

GEAR ADDENDUM.

the obliquity and formula 6 may be employed in finding the gear addendum. In.

our example however, the approach is less than the obliquity and formula 7 is used,

. (1:0.7571 Sine P=O.4292 =0.1842

we have:

and

A,= sec E d Sim. P 1.0203 x 0.7571 x 0.1842 X =01226".

The second part of formula 7 (that is, see E d sine 1 0.05) is not used in this example as it makes no material change in gear addendum for this particularly gear ratio. As miter gears are approached, however,

it must be employed.

Pin/ion addendum. The pinion addendum is found by formula 3- as follows:

A D sec E 0.7571 X 1.01

Having settled upon theangle of approach and of recess and determined therefrom the gear and pinion addenda, we next determine the proper number of teeth in contact by formula. 8 and derive the number of teeth in pinion by means of formulae 9 or 10, as

the case may be.

Number of teeth in contact.

The desired number of teeth in contact is,

according to formula 8, as follows: 40 0) N :1.4 cos A:1A 0.766-1.0

It is recommended in this connection that the value of N be held above 1.0. This means that, based upon our maximum rec- (9) n: 360 N cos E ommendations 1.6 cos A) an angle of 48 0 must not be exceeded.

Number of teeth in pinion.

The application of formulae 9 or 10 will usually result ina fractional value. The

nearest integral number is taken as the number of teeth in pinion and a check is -then made by means of a second part of formula 9 or by formula 10 in'order to see that the limits set for N by formula 8 are not ex- V ceeded.

Since the angle of approach is in this example less than the angle of obliquity, for- 1', is 14 10'. We have, therefore:

' P ZQS O 6 :21 25' B I14 10" N :1.07

COS E 209899 and below 1.0.

Accordifig to the second part of formula 9 we have:

i N P,'+B+B 58 .6

n sec E;

This comes within the minimum limits of formula 8- (l.25 0.766fl.977) but falls below the general minimum set (1.0). Still in case a full length can be assured it would be a proper value. 'To raise N as desired it will be necessary to increase the spiral angle from 40 to about 42 or increase the cutter obliquity. Suppose, however, that a full bearing is assured and that we proceed upon the riginal assumption as to'normal or cutter obliquity and spiral angle, employing 6 teeth in pinion.

Teeth in gear and pinion.

The number of teeth in the ear is, of course, the number of teeth in pinion times the ratio of reductionN=nRfl 7=42. It is recommended that even ratios, such as this be avoided, so that instead of a 6/42 ratio, a 6/41 or 6/43 ratio could be used.

Total number of teeth in contact.

It is recommended that the total number of teeth in contact (N found by formula 19, is not allowed to fall below 2.0. In order to check this, the axial circular pitch (0.) must be first found.

For this we require the pitch cone distance (e), the face (f), which is assumed to be 14', and the average circular pitch (o Usual recommendation for maximum width of face is one quarter the pitch cone distance (e).

max. f- 4 4 1.339, 1} however is used.

( c: 6.2832d 6.2832 X 0.7571 0.7941

sewl u (17) c =c e -0.7941X. 5.357 0.700

2.45 teeth total contact will be satisfactory (3.0 recommended).

Operating clearances.

( 21 bottom clearance- 0.014242% 0.0142 45.357 0.0330" (22 side clearance 0.0035 /2= 0.0035x 45.357 0.0082

Tooth thickness.

- c 0.794v (23) thIGkDGSSIOf gear tooth, t g I -02380 24) thickness of pin. tow, a,,- 0.794- 0.238 5500" 7 Our final dimensions are:

Dimensions.

Dimensions; gear and pinion.

Normal obliquity 0' Virtual obliquity 25 25' Spiral angle 0' What I claim is: f 1. The art or method of determining the moth parts of intermeshing gears waist I i in generating the teeth of the ge 1 ,asasea in accordance with the following formula.

d see E sine p%=gear" addendum and in generating the teeth of the pinion in accordance with the following formula co s p iml 7 sec E in both of which formulas pinion addendum Tan p= tan of normal obliquityX secant spiral angle Cot E= /ratio of reduction for spur gears (Jot E=ratio of reduction for bevel gears tan p S1116 P m d= pitch radius of spur or virtual radius of bevel pinion. l5 cannot exceed value of P.

(2: assumed within known limits.

2. pair of intermeshing gears having generated gear teeth in which the tooth parts of the gear are based upon the formula as follows:

sec E in both of which formulas =pinion addendum Tan p= tan of normal obliquity X secant spiral angle Cot E= /ratio of reduction for spur gears Cot E=ratio of reduction for ,bevel gears tan p sec 9 (Z= pitch radius of spur or virtual radius of hovel pinion ll cannot exceed value of p Q= assumed within limits.

Sine P In testimony whereof, I have hereunto signed my name at Syracuse, in the county of Onondaga, and State of New York, this 27th day of September 1919.

cams H. LOGUE. 

